Statistics > Methodology

Abstract: Dramatically expanded routine adoption of the Bayesian approach has
substantially increased the need to assess both the confirmatory and
contradictory information in our prior distribution with regard to the
information provided by our likelihood function. We propose a diagnostic
approach that starts with the familiar posterior matching method. For a given
likelihood model, we identify the difference in information needed to form two
likelihood functions that, when combined respectively with a given prior and a
baseline prior, will lead to the same posterior uncertainty. In cases with
independent, identically distributed samples, sample size is the natural
measure of information, and this difference can be viewed as the prior data
size $M(k)$, with regard to a likelihood function based on $k$ observations.
When there is no detectable prior-likelihood conflict relative to the baseline,
$M(k)$ is roughly constant over $k$, a constant that captures the confirmatory
information. Otherwise $M(k)$ tends to decrease with $k$ because the
contradictory prior detracts information from the likelihood function. In the
case of extreme contradiction, $M(k)/k$ will approach its lower bound $-1$,
representing a complete cancelation of prior and likelihood information due to
conflict. We also report an intriguing super-informative phenomenon where the
prior effectively gains an extra $(1+r)^{-1}$ percent of prior data size
relative to its nominal size when the prior mean coincides with the truth,
where $r$ is the percentage of the nominal prior data size relative to the
total data size underlying the posterior. We demonstrate our method via several
examples, including an application exploring the effect of immunoglobulin
levels on lupus nephritis. We also provide a theoretical foundation of our
method for virtually all likelihood-prior pairs that possess asymptotic
conjugacy.